Segre-driven radicality testing

Article Mathematics

Probabilistic Saturations and Alt's Problem

Jonathan D. Hauenstein et al.

Summary: Alt's problem is to count the number of four-bar linkages that interpolate nine general points in the plane. This problem can be solved by counting the solutions to a system of polynomial equations, but it is still unknown if all solutions have been obtained. Therefore, we consider upper bounds for Alt's problem by counting the solutions outside of the base locus to a system generated by linear combinations of polynomials. We introduce effective symbolic and numeric methods using probabilistic saturations to study such systems.

EXPERIMENTAL MATHEMATICS (2022)

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Article Computer Science, Theory & Methods

On the Complexity Exponent of Polynomial System Solving

Joris van der Hoeven et al.

Summary: The study presents a probabilistic Las Vegas algorithm for solving generic square polynomial systems over finite fields, achieving nearly quadratic running time in the number of solutions for densely represented input polynomials. Additionally, a nearly linear bit complexity bound is proven for polynomial systems with rational coefficients. These results are obtained through a combination of the Kronecker solver and a new improved algorithm for fast multivariate modular composition.

FOUNDATIONS OF COMPUTATIONAL MATHEMATICS (2021)

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Article Mathematics, Applied